This forum is dedicated to the student just starting out with the concepts of the Reciprocal System, or RS2. Questions and clarifications for the RS/RS2 concepts go here; please place new ideas and commentary in the appropriate RS2 fora.

I also think that an axiomatic approach would be best. We may save the polemics and comparison with the current theories for later, but first thing should be to understand the basics and what is it all about on purely theoretical level. And if I may suggest that concrete examples be given for illustration whenever a new statement or notion is introduced - this would make the text more understandable than if it is only general talk which is not always clear to what exactly is refering.

I completely agree with Ardavarz and a lot of illustrations is a good idea. Periodical updates are also a good idea.

Now to anyone who whants to help: can you make some site/google-group/anything where we can put all the material and more people can comment it, make illustrations etc. ? Its also needed to limit it to a small group of interested people, otherwise it will be hard to follow all the comments and changes.

Would it be correct to represent what Larson calls "progression of the natural reference system" with diagonal line in the complex plane?

That depends if the "progression of the natural reference system" is 1-dimensional or 3-dimensional. See this link. If the progression is assumed to be 1D then it begs the question "is the progression a sequence of positions along a line" ?

If "yes" then this is not thinking in terms of "speed only" but in terms of succession of positions along a line in some kind of container.

It will be very hard to draw anything before the concepts of RS2 start yielding points and positions

I think the postulates require discrete motion in three dimensions. Numerically, this progression can be expressed as:

s/t = 13/13, 23/23, 33/33, ...n3/n3

Granted, space and time do not progress as discrete cubes. However, there are definite continuous magnitudes related to these discrete magnitudes in a very fundamental fashion, that will permit conversion.

But if we ignore the discrete vs continuous issue for the moment, it's important to note a connection with Newton's third law here: Can't we can say with confidence that since the space/time expansion is in all directions equally, it must be equal in all opposing “directions” too? And since there exist 21= 2 opposing “directions” in one dimension, 22 = 4 opposing “directions” in two dimensions, and 23 = 8 opposing “directions” in three dimensions, we can write the expansion as a combination of these "directions" in n-dimensions:

1) 1(20+21+22+23)/1(20+21+22+23)

2) 2(20+21+22+23)/2(20+21+22+23)

3) 3(20+21+22+23)/3(20+21+22+23)

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n) n(20+21+22+23)/n(20+21+22+23)

This is nothing less than the expansion of Larson's 2x2x2 stack of 8 unit cubes, which 3D expansion can be expressed as a progression of 3D numbers:

23 = 8, 43 = 64, 63 = 216, 83 = 512, ...(2n)3, which we can see from above is the progression of the 3D component of the stack, as each unit of space/time increases in the progression.

Since these 3D numbers can be added, subtracted, multiplied, and divided, in what is known as a normed division algebra, with all its properties intact, this should allow for calculations of scalar motion units, combinations of units and relations between them.

If so, putting the RS on such a mathematical footing would certainly be a big boon to its ongoing investigations and studies.

You forgot to clearly differentiate between directions and "directions"

Also, no one here will have a problem with the statement that a 1D line has 2 directions, but everyone here wil balk at the notion that a 2D plane does not have infinite directions (as in the attached "pencil of lines"). You must clarify the difference between "the number of directions in a plane" and "the number of directions of a plane".

since the space/time expansion is in all directions equally, it must be equal in all opposing “directions” too?

It is unclear, what is equal to what? There should be two parameters in any such comparison statement using words like "equal", "greater" or "less".

You are right, of course. The distinction between the number of vector directions (pencil lines) in 2 or 3 dimensions is infinite, but the number of scalar "directions" in these dimensions is 2 for each dimension.

We can think of scalar "directions" in a discrete expansion/contraction as poles in a multipole expansion: monopole = point; dipole = line out (in) from (to) point in two opposed "directions;" quadrupole = plane out (in) from (to) point in four opposed "directions;" and octopole = cube out (in) from (to) point in 8 opposed "directions."

If a lower-order pole (monopole or dipole) is non-zero, then the orientation of the quadrupole in a 3D space depends on the choice of the point of view. Since there are three choices of the 1D line within the 3D stack (x, y or z axis), then there are three corresponding choices of the 2D plane within it as well (xy, xz or yz). There is only one choice for the octopole, defined by the eight opposing corners of the stack connected by diagonals through the origin (point.)

Just as the mathematical expansion of the stack creates an ever increasing volume, defined by the symmetrical increase of the number of cubes in the respective stacks (2x2x2 = 8, 4x4x4 = 64, 6x6x6 = 216, 8x8x8 = 512, ...nxnxn = (2n)3, the physical expansion in the 8 scalar "directions" of 3 dimensions defines a ball in its interior containing the infinite number of possible pencil lines, called a vector space, which corresponds to the number of cubes in the stack (the magnitude of the stack's three dipoles will be equal to the ball's diameter.)

Hestenes' geometric algebra is an algebra for defining and manipulating the possible vector directions and magnitudes within this 3D space, but I don't think there is an algebra for defining and manipulating the possible scalar "directions" and magnitudes in this space, let alone creating it in the first place, which is what we must do.

Doesn't the concept of container and the position in it appear very much later later in the development of RS2 ?

Yes, it does. Before you can extract locations and directions, you have to create the "clock," which does not exist in the natural reference system. It is a tool of projection, to create the illusion of reality.

You can still visualize the projections, though, which helps to build the connection between our conventional coordinate reference system to the natural reference system of motion.

rossum wrote: Now to anyone who whants to help: can you make some site/google-group/anything where we can put all the material and more people can comment it, make illustrations etc. ? Its also needed to limit it to a small group of interested people, otherwise it will be hard to follow all the comments and changes.

The RS2 site has all the facilities you need; even an equation editor and a book format, where you can comment on individual sections and revise as needed. When you want to get started, let me know and I'll create you a workspace. I can limit access to just specific people also.

If the scalar direction can be either inward or outward (I prefer to think about it as "hither" and "thither" instead - "in" and "out" somehow suggest moving from a volume and so - in a container, but maybe this is just me), then perhaps we can express it with binary numbers or as in the symbols of I Ching - with broken and unbroken lines. And when there is three dimensions of motion, all possible combinations can be symbolized by the eight trigrams (ba gua) or 3-digit binary numbers.

Also I think that our habit of considering space (and by extension - time) as coordinate grid of orthogonal cells (squares or cubes) is a result of cultural conditioning. It seems to us "natural" because we have been taught so and we are accustomed to it, but if space is really isotropic there can be no privileged directions of the axes and thus every orientation of the cubes (or squares) should be equally right. So maybe a better representation would be to consider space like expanding circles (resp. spheres) and thinking in terms of polar (resp. spherical) coordinates (or maybe bipolar with variable parameter of distance between every two points, thus the fundamental notion behind "spatiality" would be not the abstract "point", but rather the interval - distance, i.e. extension).

The geometry in general was developed mainly by sedentary agricultural civilizations and as its name shows (from Greek "earth-measuring") it served the purpose of measuring lands and fields on the earth surface. Then the concept of abstract space as "container" or "stage" on which the motion is performed also derives from the observation of movement upon and in respect to the earth surface. Thus our mind has got used to regard "space" as some abstract "earth" being the background of motion and likewise to fragment it in abstract "fields" just as it does with the earth. Sometimes I think that even our notion of "function" in calculus is also derived from this - it is like calculating the amount of crops (the function) from particular field (the argument) reduced to infinitesimality. But there is also another way of indicating locations in space that is more peculiar to nomadic or sailing societies and it actually uses terms of motion. Thus to define the position of one location in respect to another is to describe the path of travel from one to the other, like f.e. so-and-so time units (hours, days etc.) of travel (walking, riding or sailing) in such-and-such direction. This is more like thinking in polar coordinates than in orthogonal ones - that is in distance (expressed as time assuming some relatively constant average speed) and direction (a point of the compass, so - an angle - azimuth). As inhabitants of the mainland we are used to think about the universe like we think about the land, but it seems to me that the idea about "universe of motion" requires thinking of it rather as a sea (which evokes multitude of ancient mythological ideas about the "cosmic ocean", "celestial waters" etc.).

I am an LRC newbie myself. I do not have advanced degrees in any field of science. I can handle calculus, diff equs, abstract algebra etc. I have read a few of Larson's books and still there are 2 questions that remain unresolved. Can anyone:

1. Prove mathematically that time and space really are recirocals of one another, at least in regards to motion.

2. Prove that motion(s) gives rise to mass. If you can identify all of the motions assigned to an object and assign number values to that motion, you should be able to measure and predict what the mass of an object will be. Einstein showed that mass increases with velocity/momentum, but never touched on what created mass in the first place.

I would think if one were able to do these 2 things that even an entrenched, fundamentalist "science minded person" would have to grudgingly admit that Larson was onto something.